A094687 Convolution of Fibonacci and Jacobsthal numbers.

0, 0, 1, 2, 6, 13, 30, 64, 137, 286, 594, 1221, 2498, 5084, 10313, 20858, 42094, 84797, 170582, 342760, 688105, 1380390, 2767546, 5546037, 11109786, 22248228, 44542825, 89160674, 178442742, 357081901, 714481614, 1429477456, 2859786953

Offset

0,4

Comments

Also convolution of A008346(n-1) and A000079(n).

Also difference of Fibonacci and Jacobsthal numbers shifted left: a(n) = A000045(n+1) - A001045(n+1). - David Callan, Jul 22 2008

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,2,-3,-2).

Formula

G.f.: x^2/((1-x-x^2)*(1-x-2*x^2)).

a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) - 2*a(n-4).

a(n) = Sum_{k=0..n} A000045(k)*A001045(n-k).

a(n+1) = a(n) + 2*a(n-1) + A000045(n). - Philippe Deléham, Mar 06 2013

a(n) = J(n+1) - F(n+1) = Sum_{k=0..n} F(k)*J(n-k), where J=A001045, F=A000045. - Yuchun Ji, Mar 05 2019

Example

a(2) =   0 + 2*0  +  1 =   1
a(3) =   1 + 2*0  +  1 =   2
a(4) =   2 + 2*1  +  2 =   6
a(5) =   6 + 2*2  +  3 =  13
a(6) =  13 + 2*6  +  5 =  30
a(7) =  30 + 2*13 +  8 =  64
a(8) =  64 + 2*30 + 13 = 137
a(9) = 137 + 2*64 + 21 = 286
... - Philippe Deléham, Mar 06 2013

Maple

with(combstruct):
TSU := [T, { T = Sequence(S, card > 1), S = Sequence(U, card > 0), U = Sequence(Z, card > 1)}, unlabeled]:
seq(count(TSU, size = j+2), j=0..32); # Peter Luschny, Jan 04 2020

Mathematica

LinearRecurrence[{2, 2, -3, -2}, {0, 0, 1, 2}, 40] (* G. C. Greubel, Mar 06 2019 *)

Prog

(PARI) my(x='x+O('x^40)); concat([0, 0], Vec(x^2/((1-x-x^2)*(1-x-2*x^2)))) \\ G. C. Greubel, Mar 06 2019
(Magma) I:=[0, 0, 1, 2]; [n le 4 select I[n] else 2*Self(n-1) + 2*Self(n-2) -3*Self(n-3) -2*Self(n-4): n in [1..40]]; // G. C. Greubel, Mar 06 2019
(Sage) (x^2/((1-x-x^2)*(1-x-2*x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Mar 06 2019
(GAP) a:=[0, 0, 1, 2];; for n in [5..40] do a[n]:=2*a[n-1]+2*a[n-2] - 3*a[n-3]-2*a[n-4]; od; a; # G. C. Greubel, Mar 06 2019

Crossrefs

Sequence in context: A289048 A297388 A115217 * A369584 A336875 A219753

Adjacent sequences: A094684 A094685 A094686 * A094688 A094689 A094690

Keyword

easy,nonn

Author

Paul Barry, May 19 2004

Status

approved